Team:GENAS China/Model

Model’s purpose

1. The model aims to proof the feasibility of recombinase system as an eligible biological relay, predicts the narrow hypersensitive response range of recombinase, which is different from normal transcription factor based genetic switch and crucial for the significant state transition.

2. Providing the mathematic basis of that different recombinase and RBS can result in the change of response range of recombinase system.

Assumption

1. This model describes the reactions that happen under equilibrium condition

2. The recombination process of attP/attB sites is much slower than the binding process of recombinase

3. The concentration of recombinase won’t influence the recombinating efficiency of recombinase.

4. The amount of recombinase must be much more than the amount of attP/attB sites， thus the binding at each site is independent.

5. The binding and recombinating process at two sites are identical.

Parameter and symbol

symbol	meaning
n·Int	Polymer which include n recombinase
attB/attP	Recombinase binding sites attB and attP
n·Int·attb	Recombinase site bind by recombinase
K_b/K_p	The dissociation constant of n·Int·attb
Int	The dissociative recombinase
P_B/P_P	The probability of attB/attP being bind by recombinase
P	The probability of target DNA being recombinate
τ	The recombinating rate of certain recombinase
q	The constant ratio between the ratio between expression of GFP and recombinase, when using Ptac
α,β	Correlation factor

Modeling process

The recombination of the target DNA can break down into two independent incidents: The binding and recombination of attP and attB. Based on assumption 5, the reaction happens at attB and attP are the same. Here, we only present the detail of attB.

where the dissociation constant of n·Int·attb is K_b.

the dissociation constant can be expressed by

In order to connect the possibility and [INT], the output, we established the binding probability of attB site as a function of

.

Substitute (3) into (2) we can get a hill equation, relating the output intensity, which is represented by the concentration of recombinase and the binding probability of attB

Likewise, the binding probability of attP site can be expressed by

Because of that only when both attB and attP are combined and recombinated, the target DNA will be reversed, so the recombination probability of target DNA can be expressed by

Substituting equation (4) and (5) into (6), we get

Then we are going to relate P to the input of bio-relay. In the first and second graph, the abscissa is the concentration of recombinase. However, the amount of it can’t be directly measure. Hence, we have to relate [Int] to [GFP]([Input]).

when using the same promoter and RiboJ insulator, the ratio between two different proteins expression is constant (Lou, Stanton et al. 2012). That, if we install GFP and recombinase separately to the same promoter, their outcome intensity function will have a constant ratio. Hence, we can utilize this relationship to build a bridge between the input of the promotor and the output of the recombinase. And the ratio value between the [Input] and [Int] can be absorbed into K values. Hence, we get the equation (8).

We developed a program to simulate the transfer function with different K values (Figure 1). A very narrow hypersensitivity response interval can be observed which suggest the recombinase system can convert a small change of analog signal into a quantitative change of digitl signal which is a feature of relay device.

Figure 1. The simulation of recombinase transfer functions with different K values

Since the recombinases will form dimers before they bind to attB and attP, the n and m are equal to 2. The x-axis indicate the input signal.

It seems like the independent change of K_p and K_b can also influence the feature of the function curve (Figure 2).

Figure 2. The simulation of recombinase transfer functions with independent changed K values

According to (8), P are influenced by K_p and K_b which are depends on the recombinase itself. Thus, the alternation on the recombinases or finetuning on the recombinases' RBS sequence can influence the function.

We attempt to fit a set of normalized experimental data using this model to see if the therotical result matched the experimental result. In order to get the precise fit resluts of the key parameters, we add two correction factors α and β to overcome the deflection caused by the measure method and data procession, getting the equation (9).

Figure 3 is the fitting results of 3 integrase system with equation (9).

Figure 3. The fitting results of phiC31(C35), Int10(T25) and TG1(G25) system.

Our experiment result proof our prediction, the hypersensitive response range of the recombinase is narrow enough, meanwhile, various recombinase can alter the function and increase or decrease the response range.

This transfer function can be used for predicting the combination performance of the recombinase module with another signal source of input.
For example, a person who wants to detect [Hg⁺] from 0.01M to 0.1M should first determine the GFP expression at 0.01M of [Hg⁺] and 0.1M of [Hg⁺ ]. Then conforme if the two thresholds are situated in the two sides of the hypersensitivity response interval of the recombinase system. Hence, one can select the recombinase that has the corresponding output intensity and apply it to the design.

Consequently, we are able to depict the ability of different promotors when applying to our bio-relay through determine the function of the promotor when it’s promoting GFP. For example, the response function of a promoter used as a sensor is (10)

Because the output of the sensor is the input of the relay module, we can get the final response function of the conbination system:

[1]Lou, C., et al., Ribozyme-based insulator parts buffer synthetic circuits from genetic context. Nature Biotechnology, 2012. 30(11): p. 1137-42.
[2] Chunbo Lou, Brynne Stanton, Ying-Ja Chen, Brian Munsky & Christopher A Voigt. Ribozyme-based insulator parts buffer synthetic circuits from genetic context. Nature biotechnology.